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Turbulence, waves, and jets in a differentially heated rotating annulusexperimentR. D. Wordsworth, P. L. Read, and Y. H. Yamazaki Citation: Phys. Fluids 20, 126602 (2008); doi: 10.1063/1.2990042 View online: http://dx.doi.org/10.1063/1.2990042 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v20/i12 Published by the American Institute of Physics. Related ArticlesA characteristic function to estimate the longitudinal dispersion coefficient in surface water flows over porousmedia Phys. Fluids 24, 046602 (2012) Frequency and damping of non-axisymmetric surface oscillations of a viscous axisymmetric liquid bridge Phys. Fluids 24, 042103 (2012) Nonlinear resonance in barotropic-baroclinic transfer generated by bottom sills Phys. Fluids 24, 046601 (2012) An eddy viscosity model for two-dimensional breaking waves and its validation with laboratory experiments Phys. Fluids 24, 036601 (2012) Microdroplet oscillations during optical pulling Phys. Fluids 24, 022002 (2012) Additional information on Phys. FluidsJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

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Turbulence, waves, and jets in a differentially heated rotatingannulus experiment

R. D. Wordsworth,a� P. L. Read, and Y. H. YamazakiAtmospheric, Oceanic and Planetary Physics, Clarendon Laboratory, Department of Physics,University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom

�Received 16 March 2008; accepted 15 August 2008; published online 8 December 2008�

We report an analog laboratory study of planetary-scale turbulence and jet formation. A rotatingannulus was cooled and heated at its inner and outer walls, respectively, causing baroclinicinstability to develop in the fluid inside. At high rotation rates and low temperature differences, theflow became chaotic and ultimately fully turbulent. The inclusion of sloping top and bottomboundaries caused turbulent eddies to behave like planetary waves at large scales, and eddyinteraction with the zonal flow then led to the formation of several alternating jets at mid-depth. Thejets did not scale with the Rhines length, and spectral analysis of the flow indicated a distinctseparation between jets and eddies in wavenumber space, with direct energy transfer occurringnonlocally between them. Our results suggest that the traditional “turbulent cascade” picture ofzonal jet formation may be an inappropriate one in the geophysically important case of large-scaleflows forced by differential solar heating. © 2008 American Institute of Physics.�DOI: 10.1063/1.2990042�

I. INTRODUCTION

Turbulent motion is ubiquitous in the atmospheres of allgas giant planets and most terrestrial planets in our SolarSystem. The primary energy source for the turbulence is usu-ally either solar or internal heating; in general, planetary at-mospheres are almost always far from thermal equilibriumwith their surroundings.

On very large scales, atmospheric motion is usuallystrongly constrained by vertical stratification and the Corioliseffect, and hence is nearly geostrophic away from equatorialregions of the planet. Geostrophically turbulent flow behavesdramatically differently from the “classical” three-dimensional isotropic case,1 and its importance is such that itis intensively studied in the fluid-dynamical literature.

One of the most insightful early results on the subjectwas due to Fjortoft,2 who demonstrated that for any nonlin-ear triad interaction in two-dimensional �2D� flows, energy istransferred preferentially to smaller wavenumbers. The fun-damental basis for 2D and geostrophic “inverse energy cas-cade” theories of turbulence, this result broadly explains thetendency for atmospheric motion to be dominated by large-scale coherent structures such as the jets and vortices of thegas giants, despite the fact that at least some of the forcing islikely to be occurring on far smaller scales.3

Another constraint on planetary circulation that stronglyaffects the nature of large-scale flows is the variation inCoriolis effect with latitude or planetary �-effect. The devel-opment of geostrophic turbulence on an idealized �-planewas studied by Rhines, who predicted in a seminal paper4

that planetary wave motion arising from the �-effect wouldalter the inverse cascade of energy at low wavenumbers,causing the transfer of energy into the zonal modes. This

“Rhines effect” is frequently cited as an explanation for thezonal jets observed in the atmospheres of Jupiter and Saturn.

While insightful, Rhines’ work was essentially heuristicand could not provide a detailed dynamical explanation ofthe process. As a result, barotropic �-plane jet formation hasbeen the subject of intensive numerical study by many re-searchers �e.g., Refs. 5–7�. Notably, Vallis and Maltrud8 stud-ied decaying unforced turbulence and suggested that plan-etary wave motion should dominate inside a dumbbell-shaped region aligned along the axis of the east-westwavenumber kx in spectral space. From their simulations,they found that energy transfer slowed dramatically in thisregion, and instead proceeded mainly toward the zonalmodes. However, they also noted that there was no a priorijustification for zonal energy to peak at the Rhines scale.

More recently, Sukoriansky et al.9 produced an empiricaltheory for the spectral scaling observed in 2D simulations ofturbulent flows on the surface of a rotating sphere. Galperinet al.10 noted further properties of these idealized systems,including the importance of the large-scale damping mecha-nism in determining the steady-state jet structure. Finally, arecent experiment at the large-scale Coriolis facility11 alsoreproduced �-plane multiple jet formation in a convectivelydriven laboratory flow.

In real atmospheres, the role of vertical structure on thedynamics is generally non-negligible. In particular, directenergy input to the system can occur through baroclinic in-stability �ultimately due to differential solar heating with lati-tude�. Salmon12 argued that for baroclinically forced turbu-lence, the internal deformation radius can be treated as an“energy input scale” to the quasi-2D barotropic flow, whichotherwise behaves more or less as it would in the genericsmall-scale forcing case. In particular, a barotropic inversecascade is generally still expected to transfer kinetic energyfrom small to large scales.a�Electronic mail: [email protected].

PHYSICS OF FLUIDS 20, 126602 �2008�

1070-6631/2008/20�12�/126602/12/$23.00 © 2008 American Institute of Physics20, 126602-1


http://dx.doi.org/10.1063/1.2990042http://dx.doi.org/10.1063/1.2990042http://dx.doi.org/10.1063/1.2990042

As Salmon’s argument is based only on conservationlaws and an irreversibility principle, it makes quite unam-biguous predictions, but only about the general direction ofenergy transfer. Further theoretical progress has proved dif-ficult, not least because for many real flows the deformationradius is not much smaller than the observed jet width. Inthis situation standard cascade-type arguments, which are de-pendent on the existence of well-defined inertial ranges inspectral space, are of dubious validity.

Some numerical simulations of baroclinically forced tur-bulence have been carried out. Panetta13 simulated two-layer�-plane turbulence in a doubly periodic domain and foundthat equivalent barotropic jets with width proportional to theRhines scale formed, with the system reaching a steady stateover a time-scale longer than that predicted by any simplecombination of model parameters. However, the computa-tional power needed to simulate fully three-dimensional tur-bulent flows for long time periods means that this problemhas not been well studied, in general. In particular, largeuncertainties remain concerning the exact nature of energytransfer from baroclinic to barotropic modes, as well as thatbetween eddies and the mean zonal flow.

We used a differentially heated rotating annulus to gen-erate baroclinically forced turbulence in the laboratory. Firstdeveloped by Hide,14 the “baroclinic annulus” is intended asan extremely simple prototype of the motion of real plan-etary atmospheres. In essence, the idea behind the experi-ment is to mimic the effects of tropical heating and polarcooling on a midlatitude air mass by heating and cooling,respectively, the outer and inner walls of an annulus filledwith some working fluid. It has now been extensively studiedin a wide range of configurations—for a comprehensive re-view of important previous results, see, e.g., Ref. 15. How-ever, due to the difficulty involved in �a� pushing the systeminto a geostrophically turbulent state and �b� acquiring de-tailed data on the flow when such a state is reached, previousresearch has tended to focus on the weakly nonlinear behav-ior of the system.

Here, we use new data acquisition techniques and a largeannulus setup that allows us to investigate more stronglyturbulent flows in more detail than has previously been pos-sible in the laboratory or via numerical simulation. We focuson the steady-state properties of the system in the turbulentregime and examine both flat and sloping boundary cases.The latter case has previously been studied in detail only inthe weakly nonlinear and chaotic regimes.16 One past inves-tigation involving internally heated fluids in a slightly differ-ent setup began to show evidence of zonal jet formation athigh rotation rates.17 However, it was not possible in thatstudy to investigate the behavior of the jets in detail.

The aim of this paper, therefore, is to elucidate the dy-namics of the turbulent jet formation regime. In Sec. II wedescribe the experimental apparatus. In Sec. III an overviewof the results for both flat and sloping bottom boundary ex-periments is given. General features of the flow are discussedfirst, followed by progressively more detailed analyses of jetand eddy dynamics. Finally, in Sec. IV we discuss someconclusions from our results, including implications for tur-bulent flows observed in real planetary atmospheres.

II. APPARATUS

Our experimental setup is as follows. An annulus filledwith a working fluid is set on the center of a rotating turn-table, and its inner and outer walls are cooled and heated,respectively, in order to maintain a constant temperature dif-ference between the two �see Fig. 1�. By the thermal windequations �see, e.g., Ref. 1�, in a rotating frame this tempera-ture difference causes a vertically sheared zonal velocity pro-file. When the rotation rate of the turntable is increasedabove a certain critical value, this axisymmetric flow thenbecomes hydrodynamically unstable and can evolve into awide variety of steady or chaotic flow states. The state of thesystem is principally determined by two dimensionless num-bers: the Taylor number

Ta �4�2�b − a�5

�2d�

�2� � u�2

��2u�2, �1�

which is a ratio between Coriolis and viscous effects, and theHide number

undariesNeutrally buoyant

tracer material

Multi-level flow illumination

level 1 h = 20.7 cm

level 2 h = 17.4 cm

level 3 h = 13.8 cm

level 4 h = 10.4 cm

level 5 h = 6.8 cm

�

Water jacket maintains

radial temperature difference

�T

9 cm

30 cm

26.5cm

Removable sloping boundaries

Digital

camcorder

�

Water jack

radial temper

Perspex upper

boundary

Matt black lower

boundary and walls

26cm

FIG. 1. �Color online� Schematic of the apparatus used for all experiments.A temperature controlled water jacket was used to maintain the difference�T between inner and outer walls.

126602-2 Wordsworth, Read, and Yamazaki Phys. Fluids 20, 126602 �2008�


� ��gd�T

�2�b − a�2, �2�

which is essentially a ratio between buoyancy and Corioliseffects or “thermal Rossby number.”14 Definitions and valuesof all general experimental parameters are given in Table I.

Previous studies have shown that the transition to turbu-lent flow occurs at large Taylor and small Hide numbers. Weused an apparatus with a relatively large gap width �b−a�,which at high rotation rates allowed us to reach Taylor num-bers an order of magnitude greater than had previously beenused in experimental studies with sloping boundaries.17

As mentioned in Sec. I, the top and bottom boundaries ofthe annulus can be either flat or sloping. In the sloping case,motion of fluid columns in the radial direction is constraineddue to conservation of local angular momentum. For a baro-tropic fluid �no variation in velocity with depth�, the effectsof sloping topography are the same as those of the planetary�-effect. A “topographic beta parameter”17 can be defined

� =4� tan �

d̄, �3�

where d̄ is the mean depth of the fluid with boundaries in-cluded and � is the angle of the slope. Note that for simplic-ity, we use the maximum depth d in all other dimensionlessnumbers, in order to make direct comparison between flatand sloping experiments easier.

For a baroclinic �vertically sheared� flow, the analogy isno longer exact, as sloping boundaries also modify the zon-ally symmetric temperature field and hence the growth rateof instabilities. However, previous studies16 have shown thatthe qualitative aspects of flow behavior are not strongly af-fected by this change.

The internal deformation radius is another importantquantity in baroclinically forced turbulence, as it is the scaleat which energy is optimally transferred from baroclinic tobarotropic modes. Here it is defined as

Ld =��gd�Tz

�, �4�

where �Tz is the vertical temperature difference. Althoughwe do not have temperature information for these experi-ments, an upper bound on the deformation radius LmaxLdcan be derived by replacing �Tz in Eq. �4� with the imposedhorizontal temperature gradient �T. As will be seen, this

upper limit is extremely useful for analyzing the propertiesof the turbulent flow regimes.

We used flow visualization to acquire information on theinterior dynamics of the fluid. Neutrally buoyant tracer par-ticles of radius of 350–500 m were suspended in the fluidand alternately illuminated by thin ��0.5 cm� light sheets atfive different depths. For most experiments we chose a82.5%–17.5% water/glycerol mix �fluid A� as the workingfluid, in order to match the density of the tracer particles,�=1.043 g cm−3. Some experiments were also performedwith a NaCl salt solution �fluid B�: in this case the densitywas the same, but the fluid viscosity was only �=1.11�10−6 m2 s−1, allowing a greater Taylor number and hencea potentially more turbulent flow at the same rotation speedas in the previous case.

The flow was imaged from above by a digital camcorder�720�576 resolution�, which was connected to a small com-puter placed on the turntable during experiments. The turn-table computer was then controlled remotely via a wirelessethernet connection with a second computer in the laboratoryframe, allowing data acquisition and control without the needfor slip rings or similar mechanical methods.

Raw images of the flow were converted to velocity fieldsby use of correlation imaging velocimetry software �CIV�.18

In brief, CIV works by comparing images of the flow witha given time separation, tracking the displacement, rotation,and shear of image texture within small �typically10�10 pixel� boxes. Unlike more traditional particle imagevelocimetry approaches, it does not need to resolve the mo-tion of individual tracer particles, and hence is well suited tothe analysis of flows with a large degree of spatial scalevariation. For all data sets reported here, the Cartesian gridsproduced by CIV were interpolated onto 24�144 �r ,�� polargrids before analysis.

The digital camcorder used a MINIDV format, whichcompresses images before storing them. We tested the errorintroduced by this by taking raw demonstration flow imagesfrom the CIV website, http://www.civproject.org, and artifi-cially compressing them using the same algorithm as is usedfor the MINIDV video format. Both original and compressedimages were then processed in CIV, and the resulting velocityfields compared. The difference in total kinetic energy be-tween the two was less than 1%. However, fluctuations dueto image compression error tended to increase with wave-number, reaching 10% at ke=0.1 rad pixel

−1. In the spectralanalysis of Sec. III B, therefore, we limit our diagnostics tomodes with wavenumber smaller than this value.

In the case of sloping top and bottom boundaries, theslope on the top perspex boundary causes a slight deviationof light rays traveling from the working fluid to the cam-corder because of the differing refractive indices of air, per-spex, and the working fluid. To counteract this effect, theperspex boundary also had a slight slope ��top=3.5°� on itsupper surface. We also checked view distortion by photo-graphing concentric black and white rings in the tank at dif-ferent depths and found that the maximum error at the innerand outer walls was of order a few pixels only. This wasjudged sufficiently small to be neglected in the analysis.

We alternated the illumination of the five depth levels

TABLE I. General experimental parameters.

Radius of inner cylinder a 4.5 cm

Radius of outer cylinder b 14.3 cm

Annulus depth �flat� d 26 cm

Annulus mean depth �sloping� d̄ 21.5 cmGravitational acceleration g 9.81 m s−2

Sloping boundary angle � 22°

Kinematic viscosity of fluid A � 2.04�10−6 m2 s−1

Kinematic viscosity of fluid B � 1.11�10−6 m2 s−1

Volumetric expansion coefficient � 3.16�10−4 K−1

126602-3 Turbulence, waves, and jets in a differentially heated experiment Phys. Fluids 20, 126602 �2008�


using a simple oscillation circuit with a manually adjustabletime delay. This allowed us to derive quasisimultaneous hori-zontal velocity fields for the fluid at different depths for awide range of flow regimes. The time delay between levelsin the experiments was between 3 and 6.5 s, depending onthe observed flow velocity. The delay was chosen to be longenough to allow high quality CIV analysis, but short enoughto allow quasisimultaneous visualization of the flow field. Inall cases, it was smaller than �a� the minimum eddy turnovertime and �b� the period of the fastest observed planetarywaves �see Sec. III B�.

To calculate barotropic modes for the spectral calcula-tions, we interpolated multilevel data linearly in time be-tween frames before vertically averaging. For a fully rigor-ous modal decomposition, weighting by level is required dueto the variation in the interior temperature gradient T��z�with depth. However, even for unrealistically high verticaltemperature changes, this weighting deviates only by an ex-tremely small amount from unity, and the error due to thisapproximation was neglected.

Spectra were also produced from mid-depth data andcompared with the vertically averaged velocity fields. Thedifference between the two was small, suggesting that mid-depth fields were a good proxy for the interpolated barotro-pic data. We use the vertically averaged fields in Sec. III B.Finally, the scientific software MATLAB was used for all fur-ther data analysis and diagnostics. Where necessary, the de-tails of more complicated analyses performed are discussedin the next section.

III. RESULTS

As mentioned in Sec. II, the experiments presented hereinvestigate a parameter space region of high Taylor and lowHide number. The locations in parameter space of all experi-ments performed are shown in Fig. 2. Also shown, for com-parison, is a regime diagram from a previous investigation ina smaller apparatus with similar aspect and radius ratios.19 In

this investigation, we varied rotation rate and temperaturedifference from �=0.65 to 3.9 rad s−1 and �T=1 to 4 K.

As the principal interest of this investigation was the�statistically� steady-state properties of the flow, each ex-periment was run for 2 h, with data collection occurring inthe second hour only. The Ekman spinup time, defined as

Ek=d /�2��, varies between 160 and 65 s for the experi-ments presented here: hence between 22 and 55 spinup timespassed before data collection. Furthermore, after CIV analysiswe plotted total energy and enstrophy of the flow as a func-tion of time for each data set, and found that in all cases,these quantities did not exhibit any significant monotonictrend over the observed period.

Figure 3 shows streak images of the flow for flat andsloping boundary experiments at �a� low and �b� high rota-tion rates. Each image was taken after approximately 30 minof evolution time, with an averaging time of 20 s.

The streak images in Fig. 3�a� are from experimentswhere the rotation rate was low enough for the flow to bechaotic rather than fully turbulent, at least in the flat bound-ary case. There, the flow appears to have been in a structur-ally vacillating “wavenumber 4” state �4SV�, which is ex-pected, given the parameter space diagram recorded in Ref.19 �Fig. 2�. In the sloping case, a wavy jet close to the outerboundary is apparent, with the inner half of the channeldominated by a chain of moving vortices.

The streak images in Fig. 3�b� show more complicatedbehavior. The flows in both experiments were fluctuatingfairly rapidly: the flat boundary case, in particular, exhibitsvarying radial and azimuthal eddy motion, with no singlewavenumber dominating. In the sloping case, the large-scaledomain-spanning eddies were replaced by apparent wavelikemotion and weak zonal jets. The bottom-right image in Fig. 3is somewhat reminiscent of the streak images reported pre-

FIG. 2. �Color online� Logarithmic parameter space diagram showing thepositions of all experiments performed as a function of Taylor and Hidenumber �T vs ��. Also shown in black is a regime diagram derived from aprevious experimental study �see Ref. 19, for details�.

FIG. 3. Streak image comparison of the mid-depth flow in flat �left� andsloping �right� boundary cases for �a� low rotation rate �=1.3 rad s−1 and�b� high rotation rate �=3.9 rad s−1 with �T=2 K.



viously in a quite different setup;20 there, a rotating parabolicdish was heated from below and allowed to cool convec-tively at its upper surface. However, little quantitative veloc-ity field information was derived for that experiment, makingfurther comparison difficult.

Figure 4 shows CIV-derived contour plots of near-instantaneous vertical vorticity component �=r−1��r�ru��−��ur� at three different depths in the fluid for the sameexperiments as displayed in Fig. 3�b�. Three-dimensional vi-sualization of this type helps to highlight the dramatic differ-ence between the flow in the flat and sloping boundarycases—note, in particular, the pronounced wavelike appear-ance of Fig. 4�b�. Animations of the vorticity field showedthat the wave crests were traveling westward, as expected fordisturbances that are qualitatively similar to atmosphericplanetary waves. In Sec. III B, we discuss a temporal spectralanalysis of the velocity field data that shows that much ofthis flow was indeed dominated by wavelike motion.

A selection of relevant length scales as a function ofTaylor number for all the sloping boundary experiments isplotted in Fig. 5. The Rhines scale, defined as LRh=�U /�,was calculated using a time and volume averaged value forthe root mean square flow speed U. The jet scale was esti-mated by Fourier transforming the zonal, temporal, and ver-tical means of the azimuthal velocity �u��z in radius. The

location of the peak of the resulting power spectrum wasthen taken to be the characteristic jet wavenumber, with jetscale the inverse Ljet=� /kjet. This analysis was performed onboth midlevel and vertically averaged data, but the differencebetween the two was found to be extremely small.

The horizontal buoyancy scale �upper limit on deforma-tion radius� decreased with Taylor number, becoming lessthan one-tenth of the channel width for the highest rotationrate experiments. The Rhines scale for the sloping boundaryruns also decreased with Taylor number, primarily because �increased with �. Neither length scales were good quantita-tive predictors for the observed jet width, although wenote that for Taylor numbers greater than T�2�108 for the�T=2 K runs, all three scales were at least decreasing in thesame direction.

In the �T=1 K experiments, the observed flow wasgenerally too weak to allow accurate velocity field measure-ment. In the �T=4 K experiments, we observed single east-ward jets only, quite possibly because the deformation radiuswas too large to allow multiple jet formation. From here,therefore, we focus our analysis on the dynamically mostinteresting �T=2 K cases.

In Fig. 6, the instantaneous mid-depth vorticity �left� andzonal velocity u�=r�̇ �right� after 1.5 h are plotted for arange of such experiments. In the high rotation flat boundaryexperiment �Fig. 6�a�� both fields appear unstructured, withno evidence of coherent zonal structure formation. In thesloping boundary cases, jet formation at high rotation rates isclearly visible in the zonal velocity plots; the “fluid B”experiment, in particular, has a clear 2–3 jet structure

FIG. 4. �Color online� Multilevel snapshot of vertical vorticity component atlevels 1, 3, and 5, time t=3600 s for �a� flat boundary and �b� slopingboundary experiments with �T=2 K and �=3.9 rad s−1.

107

108

109

0

0.5

1

107

108

109

0

0.5

1

107

108

109

0

0.5

1

Taylor Number

Rhines scale

Deformation scale

Jet width

a)

b)

c)

FIG. 5. Relevant length scales in units of channel width �b−a� as a functionof Taylor number for �a� �T=1 K, �b� �T=2 K, and �c� �T=4 K slopingboundary runs.



�Fig. 6�d��. It is most interesting that regions where the jetsare strong tend to correspond to those where wave activity,as visible in the instantaneous vorticity fields, is highest.Also clearly apparent is the increase in wavenumber �bothazimuthal and radial� of the vorticity fields with rotation. Weexamine these issues further in the next two sections.

A. Eddy-mean flow interaction

The effect of eddies, be they wavelike or turbulent, onzonal mean quantities, can be written in terms of the aver-aged correlation of the various eddy fields. In this section weexamine the eddy �angular� momentum and heat fluxes,�ur�u�� and �ur�T�, and compare them to the observed time-averaged zonal flow profiles.

Temperature information was derived in an indirect wayvia the quasigeostrophic approximation. First, streamfunc-tion �� was derived numerically from the eddy vorticity

fields by computing the inverse Laplacian ��=�−2�� withboundary conditions �� r=a,b=0. The calculation was per-formed using standard matrix inversion algorithms from theMATLAB software package.

By the definition of geostrophic streamfunction�� p� /2��0, the hydrostatic approximation dp�=−g��dzand the linearized temperature-density relationship��=−�0�T�, the approximate relationship between eddy tem-perature and streamfunction can be written as

T� = +2�

g�

��

�z. �5�

For this analysis, Eq. �5� was converted to finite differenceform, and T� estimated at intermediate depths from themultilevel streamfunction data. The eddy heat flux �ur�T�was then evaluated at mid-depth via interpolation. In Fig. 7,time and zonally averaged eddy heat and momentum fluxesat mid-depth are plotted beside the time-averaged zonal flow.In all cases shown, the vertically averaged quantities weresimilar to those at mid-depth.

First, note that for all experiments �ur�T� is negative,indicating that the eddies are always transporting heat from

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FIG. 6. �Color online� Snapshots of mid-depth vorticity and zonal velocityfor experiments with constant temperature difference �T=2 K. The casesare �a� flat boundary �=3.9 rad s−1, ��b� and �c�� sloping boundary�=1.3,3.9 rad s−1, and �d� same rotation rate as �c� but with low viscosityworking fluid B.

FIG. 7. �Color online� Time and zonal averaged zonal velocities ū�, eddymomentum flux �ur�u�� center�, and approximate eddy heat flux �ur�T��right� for the same experiments as in Fig. 6.



larger to smaller radii. This is expected as the radial tempera-ture gradient is the only energy source in the experiment.Indeed, in a sense the differentially heated rotating annuluscan be regarded as a heat engine, creating jet and eddy mo-tion through the conversion of thermal to kinetic energy.

In Figs. 7�b�–7�d�, the eddy momentum flux �ur�u��is converging onto same-sign zonal flow, indicating that ed-dies were forcing the jet in each case. However, in Fig. 7�a��the flat boundary case� positive eddy momentum flux is con-verging on a negative zonal jet. This is an indication that thezonal flow may have been losing kinetic energy to the eddiesthere.

We analyzed the Rayleigh–Kuo criterion for barotropicinstability, �−�r�r−1�r�ru��z��0, with u��z the zonal andvertical average of u�, for all experiments, and found that itwas usually satisfied in the high rotation rate flat boundaryexperiments, but never in the sloping ones. In Fig. 7�a�, thepositive momentum flux convergence was therefore mostlikely due to barotropic �or mixed barotropic/baroclinic� in-stability of the negative zonal flow.

In contrast to the jets produced in this experiment, thosein the gas giant planets and also those produced in somerecent large-scale convection experiments21 have been ob-served to persistently break the Rayleigh–Kuo stability crite-rion. Galperin et al.10 discussed the difference between stableand unstable flow regimes within the context of purely 2Dbarotropic theory. In the stable case, which they called fric-tionally dominated flow, they argued that Ekman and viscousdamping act to damp the jets before they can intensifyenough to break the Rayleigh–Kuo criterion.

More sophisticated eddy-mean analyses of flows withnontrivial vertical structure take eddy heat effects into ac-count via the Eliassen–Palm �EP� flux formalism.22 In acylindrical coordinate system, with increasing radius equiva-lent to a “southward” direction, the EP flux divergence isdefined as

�m · F = r−1�r�rFr� + �zFz

= − r−1�r�r�ur�u�� − �z��f0/dzT0��ur�T�� . �6�

In the quasigeostrophic limit, Eq. �6� is equivalent to theradial flux of eddy potential vorticity q�,

�m · F = �ur�q� . �7�

As has been mentioned, it was not possible to collect tem-perature information directly for these experiments, so herethe vertical temperature gradient dzT0 is replaced with the�larger� approximate value �T /d. The vertical divergencecomponent �zFz is therefore likely to be slightly underesti-mated in this analysis.

In Fig. 8, the time and zonal averaged EP flux vector Fin two experiments is plotted as a function of r and z, withthe zonal mean profile ū� superimposed. In Fig. 9, the timeand zonal averaged EP flux divergence componentsr−1�r�rFr� and �zFz are plotted separately, for the same ex-periments as in Fig. 9. If the zonal flow is entirely

maintained by eddy motion, it is expected that ū� and �m ·Fwill be of the same sign in all regions. Conversely, if eddiesare acting to weaken the zonal flow anywhere, the two quan-tities will be of opposing sign.

� ��

��

��

��

��

��

��

��

� � �

�

�

�

�

�

�

� ��

��

��

��

��

��

�

��

��

� �

�

�

�

�

a) b)

〈uθ〉[cm][s−1] 〈uθ〉[cm][s−1]

FIG. 8. Time and zonally averaged zonal velocities and EP flux vectors forthe experiments corresponding to Figs. 6�a� and 6�c�.

� ��

��

�

�

��

��

�

�

� � �

�

�

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��

� ��

��

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�

��

� ��

�

��

��

�

�

��

��

b)

∇ · F|z × 102∇ · F|z × 101� ��

��

��

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��

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��

��

� � �

�

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� ��

�

��

��

�

�

��

��

a)

∇ · F|r × 104 ∇ · F|r × 104

FIG. 9. Time and zonally averaged EP flux divergence components for theexperiments corresponding to Figs. 6�a� and 6�c�. Note that as � ·F zinvolves a double derivative in the vertical direction, it could only be cal-culated for three different depths.



The magnitude of the z-component is considerablygreater than that of the r-component in both the examplesshown. It therefore appears most likely that heat effects wereof greater importance to the steady-state zonal flow. Notethat this was also found to be the case in some previousstudies; for example, Ref. 23, in which an internally heatedflat boundary numerical experiment was performed at lowerTaylor numbers.

The qualitative features of the divergence fields are alsoof interest. In the sloping boundary plot �Fig. 9�b��, there is aclear correlation in places between u� and both EP diver-gence components, r−1�r�rFr� and �zFz. This implies thatboth heat and momentum eddy fluxes were acting to main-tain features of the observed zonal flow field.

In the flat boundary experiment, u� was mostly anticor-related with r−1�r�rFr�. This implies that eddies were weak-ening the zonal flow, most likely through a combination ofbaroclinic and barotropic instability. The �zFz field is moredifficult to interpret, as it does not clearly relate to either thezonal flow or the eddy momentum flux divergence field. It isnegative almost everywhere, peaking in the lower half of theplot. In the flat boundary experiments at lower rotation rates,we found that �zFz anticorrelated with the zonal flow, imply-ing that the initial unstable zonal flow profile was beingweakened by baroclinic instability. For all high rotation ratecases, however, the �zFz field took the general form seen inFig. 9�b�. It is possible that the change in �zFz is linked to thetransition to turbulence, in which the vertically sheared zonalflow becomes increasingly barotropic. However, without awider range of cases to study, we could not pinpoint thephysical mechanism behind the change.

As the time-averaged zonal acceleration �t�ū� is ex-pected to be small, �m ·F was most likely balanced by othereffects in the experiments. In the sloping case, where corre-lation of �ū� with the EP flux implied eddy forcing, Ekmanand direct viscous damping were probably the main effectsacting to keep the zonal flow roughly constant. In the flatcase, as eddies were generally acting to weaken the zonalflow, it was probably deriving most of its energy directlyfrom the thermal forcing. However, it is possible that bound-ary layer effects also played a role in the overall energybudget. As we have mentioned, the vertical component of theEP divergence in the flat boundary case is rather unusual, andprobably deserves further investigation.

B. Spectra

In any turbulent or nonlinear flow, the spectral view canprovide great insight into the nature of the underlying dy-namics. In this section, we examine 2D vertically averagedenergy spectra for the sloping and flat boundary experiments.

Spectral analysis of an annular flow is more complicatedthan the rectangular channel flow case, as the correct eigen-mode expansion requires combinations of Fourier and Besselmodes. It can be shown24 that any scalar � defined in a 2Dannular domain b�r�a, with boundary conditions of theform ��a ,��=��b ,��=0, can be expanded in terms of thecomplete orthogonal basis set

��r,�� = �m=−�

m=+�

�n=1

n=�

�amnJm��mnr� + bmnYm��mnr��eim�, �8�

where Jm and Ym are Bessel functions of the first and secondkinds and �mn is a constant that can be determined numeri-cally. For this analysis, we used a standard fast Fourier rou-tine to perform the azimuthal spectral transform and then asemianalytical method to derive the radial basis modes forthe laboratory annulus.24 Numerical linear algebra routinesthen projected experimental data onto the radial modes.

As the observed flows were, in general, quite aniso-tropic, it is of most interest to examine the spectra in 2Dwavenumber space. In Fig. 10 �left column�, we have plottedtime-averaged barotropic energy spectra for several experi-ments as a function of azimuthal and radial wavenumbers mand n.

In the flat boundary case, Fig. 10�a�, kinetic energy de-creases rapidly with wavenumber, although there are smallpeaks of energy at m ,n= �4,1�. In the other plots, which areall from sloping boundary experiments, the concentration ofenergy in the �0,1–3� zonal modes is always apparent. How-ever, there are also peaks of energy at higher azimuthalwavenumber �approximately m=10, 15, and 16 for Figs.10�b�–10�d��. These peaks are evidently due to the travelingwave structures seen in the midlevel vorticity plots.

We investigated the behavior of these waves further byFourier transforming the relevant vorticity modes from each

�

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�

�

�

a)

b)

c)

d)

FIG. 10. �Color online� �Left� Time-averaged 2D contour plots of barotropickinetic energy in m ,n spectral space for the same experiments as in Fig. 6.�Right� Temporal Fourier power spectra of spectral vorticity �mn for themodes marked by an asterisk on the contour plots.



experiment in time. The chosen modes are tagged with anasterisk � �� in each case, and the resulting �normalized�power spectra �mn��2 /��mn��2d� are plotted in Fig. 10alongside the 2D spatial energy spectra.

While the flat boundary case has a fairly broadband dis-tribution of frequencies at ��15,2�, centered around �=0,the negative frequency peaks for the sloping experiments areclearly apparent. Initially, we calculated barotropic planetarywave frequencies,

� =− �k

k2 + l2, k =

m

�b + a�/2, l =

�

b − a, �9�

and compared them with the peak locations. We found thatEq. �9� significantly overestimated the magnitude of the ob-served values in every case studied.

Next, wave frequencies were calculated using the moregeneral linear instability analysis of Hide and Mason,16

which includes baroclinic and Ekman effects. The predic-tions are indicated by the bars on the plots in Fig. 10 for themean vertical temperature difference range 0.2�T��Tz�0.5�T, in order to give a rough guide to the true analyticalprediction.

Although the Hide–Mason prediction is much more ac-curate than the barotropic one, it still overestimated the wavefrequency slightly, even with the vertical temperature gradi-ent uncertainty taken into account. Note that it neglects theeffects of annulus curvature. It is not known to what extentthis affects accuracy, although errors due to other approxi-mations �such as the neglect of direct viscous damping andEkman layer heat transport� may well have been moresignificant.

At high rotation rates, the peaks become broader andmore structured �Figs. 10�c� and 10�d��, almost certainly dueto nonlinear mode-mode coupling. Interestingly, qualitativelysimilar peak broadening and shifting due to nonlinear effectshave been observed by Sukoriansky et al.25 in a numericalstudy of purely 2D planetary wave turbulence on the surfaceof a sphere. In the next section, we examine the exact natureof the interaction between modes using a spectral transfercalculation.

C. Spectral energy transfer

As a final investigation into the dynamics of the flow, wecalculate the transfer rate of kinetic energy in spectral space.This can be seen as a complement to the EP flux divergenceanalysis carried out in Sec. III A, as our main interest isagain the transfer of energy between eddies and zonal flow.However, vertically varying effects are ignored in this sec-tion. Instead, the emphasis is now on determining the extentto which energy exchange between modes is nonlocal inspectral space. Here, we assume an exchange is nonlocal if itresults in a direct net transfer of energy between two modesthat are not adjacent in spectral space.

To keep things simple, we only examine spectral transferin the azimuthal direction. This removes the need to calculatetriad coefficients involving Bessel functions, which simpli-fies the algebra considerably. Given a quasigeostrophic inte-rior flow, the quantity

Pm = − �m=p+q

�Tmpq �10�

is the time and radius averaged rate of energy transfer intoazimuthal mode m due to all barotropic nonlinear interac-tions. In a steady-state system, it must be balanced by non-conservative effects such as Ekman damping or by nonlinearinteractions of another type, e.g., mixed barotropic-baroclinic. The result �10� is derived in full in the Appendix,where the explicit form for the spectral transfer term Tmpq isalso given. Our approach was partly inspired by Ref. 6 inwhich the spectral transfer from large to small wavenumbersin a numerical simulation of 2D �-plane turbulence wasstudied.

The sum on the right hand side of Eq. �10� is the familiarone over all triads satisfying m= p+q. It is too complex andtime consuming to study the set of all possible triad interac-tions, so we chose to examine Eq. �10� for certain restrictedsubsets of wavenumbers. First, we focus on a subset of in-teractions that can cause nonlocal transfer of energy.

In Fig. 11, Pm is shown for the domain −15�m�15,but with p and q restricted to the wavenumbers marked ingray for each plot. In short, Fig. 11 shows us the energy inputto the entire range of wavenumbers between −15 and 15from the wavenumbers marked in gray in each case. By defi-nition, energy transfer to modes that are well outside the grayregions on the plots will be nonlocal. The values in the plotsin Figs. 11�a� and 11�b� were normalized by constantfactors for display purposes. These factors were 3.1�102

�Fig. 11�a�; flat boundaries� and 2.3�104 �Fig. 11�b�; slop-ing boundaries�. Hence the rate of energy transfer in thelatter experiments was smaller by a factor of about 102. Thisdifference is likely to have been due to the effects of plan-etary wave motion in the sloping boundary system.8

In the flat boundary case, the energy transfer did notappear strongly nonlocal, broadly speaking. The set of triadinteractions analyzed was causing intermediate wavenum-

FIG. 11. Normalized azimuthal spectral energy transfer Pm as a function ofazimuthal wavenumber for the experiments corresponding to Fig. 6�a� �flatboundaries� and Fig. 6�c� �sloping boundaries�. For each plot, the areasmarked in gray are those from which the wavenumbers p, q were selected.



bers to lose energy to low ones, while the higher wavenum-bers played a much lesser role in the exchange. In detail,however, the picture was quite complex, with significant en-ergy exchange also occurring between the lowest azimuthalmodes �top left�. In addition, the highest wavenumbers weretransferring some of their energy directly and nonlocally tothe lowest ones.

The sloping boundary experiments exhibited qualita-tively different behaviors. In almost every case, energy trans-fer to the zonal m=0 mode was found to be dominant. Forthe experiment shown �same rotation rate and temperaturedifference as flat case�, substantial zonal interaction occurredeven for high azimuthal wavenumbers, although we note thatfor the equivalent fluid B experiment �not shown�, the totalmagnitude of energy transfer decreased more rapidly withwavenumber. In all sloping cases examined, however, the netenergy transfer to the zonal mode for this subset of triadswas maximum at all wavenumbers.

We also examined the energy transfer due to local triadinteractions only. We defined a local triad interaction to beone in which the recipient mode m is adjacent to one of theother modes �p or q�. Because all interactions must satisfythe wavenumber rule, any local transfer of energy from, say,p to m= p�1 must involve a third wavenumber q= �1.

In Fig. 12, we have plotted Pm for the subset of triadinteractions where m= p�1 or q�1. Both plots were nor-malized by the same factors as in Fig. 12 �3.1�102 and2.3�104 for Figs. 12�a� and 12�b�, respectively�. As can beseen, the magnitude of the local transfer is fairly small com-pared to that of the nonlocal transfer displayed in Fig. 11.Only near m=4 and m=0 in the flat and sloping cases, re-spectively, is the magnitude of Pm significantly greater thanzero in Fig. 12.

In itself, this result does not demonstrate that local inter-action was negligible in the experimental system. It is ofcourse possible that some modes were interacting stronglywith their neighbors, but without net gain or loss of energy.Indeed, this is exactly what occurs in the inertial range ofclassical three-dimensional turbulence. However, an upscaletransfer of energy due to purely local interaction must finishat a given wavenumber, where a net transfer of energy willbe visible. The fact that Pm is small even at low wavenum-bers in Fig. 12 suggests that local interactions were inputtingonly a small amount of energy into the zonal flow relative tothe nonlocal effects shown in Fig. 11. In addition, the nettransfer was negative in adjacent modes, but not at higherwavenumbers �as might be expected for a turbulent cascade�.

Hence, it is likely that nonlocal spectral interaction had amore important role in the overall dynamics of the system.

It is also possible that significant local energy exchangewas occurring in the barotropic-baroclinic interactions,which we have not investigated here. This seems unlikely,however, given that energy exchange between barotropic andbaroclinic modes is generally most effective at the deforma-tion wavenumber kD.

1 As mentioned in Sec. I, it is generallyargued that in a baroclinically forced turbulent flow, interac-tions between barotropic modes will dominate at lowerwavenumbers. As significant barotropic interaction betweenlow wavenumbers was not observed in the sloping boundaryexperiments, it is therefore likely that energy input to thezonal flow from baroclinic eddy effects was at least as non-local as in the barotropic case.

IV. CONCLUSIONS

We investigated the behavior of a simple laboratorymodel of global atmospheric flow, the differentially heatedrotating annulus, at high Taylor and low Hide numbers.When the top and bottom boundaries were flat, we found, asin previous studies, that the locally smooth flow observed atrelatively low Taylor numbers evolved into a rapidly varyingturbulent one as the rotation rate increased.

When sloping top and bottom boundaries were present,multiple jets formed at mid-depth in the fluid. An EP fluxanalysis showed that eddies were directly feeding momen-tum into the jets. Also, spectral analysis of the barotropicmode showed that eddy energy was concentrated around adefinite peak wavenumber and frequency, even in the mul-tiple jet formation regime.

The most important result of this paper, obtained throughspectral transfer calculations, is that the eddies were ex-changing energy directly and nonlocally in spectral spacewith the zonal modes. There was also a net transfer of energydue to local triad interactions, but it was weaker than thenonlocal transfer. This suggests that turbulent cascade theory,which postulates that energy exchange between local wave-numbers will dominate, is not applicable to our slopingboundary results.

We argue that nonlocal zonal flow interaction with thewaves is likely to be the primary cause of jet formation in thesloping boundary experiments, but triad interactions involv-ing infinitesimal planetary waves and a zonal flow are well-known to be incapable of transferring energy to the zonalflow.26 It is likely, therefore, that the finite-amplitude natureof these waves is an important part of the system dynamics.Higher-order interactions of planetary waves with zonalflows have been studied by Newell27 among others, whofound that energy transfer to the zonal flow is possible, buton a slower dynamical timescale. Wave-mean flow interac-tion theory, which in its most generalized form allows wavesto be of arbitrary amplitude, also provides a frameworkwithin which jet formation by planetary wave forcing can beexplained.

FIG. 12. Normalized azimuthal spectral energy transfer Pm as a function ofazimuthal wavenumber for the experiments corresponding to Fig. 6�a� �flatboundaries� and Fig. 6�c� �sloping boundaries�. For both plots, the wave-number sum has been restricted to “local” triads only, as described in thetext.



It is of much interest to compare our results with thedynamics of Earth’s atmosphere at midlatitudes. There, theeastward Atlantic jet stream is known to be partially main-tained by a combination of eddy heat and momentumfluxes,28 much like the jet produced in our sloping boundaryexperiments. Typically, the deformation radius in the midlati-tude troposphere is of order of 1000 km,1 and the width ofthe jet stream is approximately 5000 km. In our high rotationrate sloping boundary experiments, the horizontal buoyancyscale �4� was approximately one-tenth of the channel width.As with any laboratory study, the Ekman damping was ofcourse greater. However, it has been suggested in some re-cent general circulation model studies that mean flow–eddyinteractions dominate even in the Earth’s atmosphere: see,for example, Refs. 29 and 30.

It is also interesting to compare these results with thezonation observed in gas giant planets. The differences in thegas giant case are somewhat greater, as interior convection,rather than baroclinic instability, may well be the dominantforcing mechanism there. Also, the observed Jovian jets arepersistently barotropically unstable, whereas the ones pro-duced by this experiment were stable. However, the basicmechanism of multiple jet formation due to correlated mo-tion of the smaller-scale eddies does appear to be the same inboth cases.31 Whether or not a spectrally local energy cas-cade from eddy to jet scales is occurring on the gas giantplanets is, to our knowledge, still an open question.

There are several obvious possible extensions to thework presented here. One would be a more general study ofgeophysically relevant baroclinically forced jet regimes, witha particular focus on the nature of energy transfer betweenzonal and eddy modes.32 Such a study could be performedeither experimentally or using a fully nonlinear numericalsimulation. In the experimental case, acquisition of fullthree-dimensional temperature fields would be a significantadvance, as it could lead to a better understanding of thethermal contribution to the EP flux �particularly in the flatboundary case�.

It would also be most interesting, and perhaps simpler, toattempt to simulate the results of these experiments with areduced theoretical or numerical model. If the dominant in-teraction in the sloping boundary case was between the zonalflow and planetary waves, it would appear logical in such adescription to neglect wave-wave interactions as a first ap-proximation. It is possible that a simple model based on thisidea could captures the main features of the flow rather well.

ACKNOWLEDGMENTS

We thank Professor G. Vallis and a second anonymousreviewer for their insightful and constructive comments onan earlier draft of this paper. Data acquisition was performedwith the use of CIV software, which is freely available onlineat http://www.civproject.org. This work was funded by aNatural Environment Research Council �NERC� studentship.

APPENDIX: DERIVATION OF THE ONE-DIMENSIONALSPECTRAL ENERGY TRANSFER EQUATION

Given the quasigeostrophic approximation,1 the momen-tum equation in the absence of forcing and damping issimply

Du = �tu + u · �u = − f � ua, �A1�

where u is geostrophic velocity, ua is ageostrophic velocity,and u only has components in the horizontal plane.

We are interested in barotropic nonlinear interactionsand so ignore the right hand side of Eq. �A1�. In cylindricalcoordinates the remainder then becomes

��t

+ ur�

�r+

u�r

�

��ur − u�2r = 0,

�A2�

��t

+ ur�

�r+

u�r

�

��u� + uru�r = 0.

We define all variables in terms of their azimuthal Fouriercoefficients,

u��r,�,t� = �m

eim�u�,m�r,t� ,

�A3�ur�r,�,t� = �

m

eim�ur,m�r,t� .

Then substitution of Eq. �A3� into Eq. �A2� followed by aFourier transform of the entire expression leads to

�ur,m�t

+ �m=p+q

apq = 0,

�A4��u�,m

�t+ �

m=p+qbpq = 0,

with the terms apq, bpq defined as

apq�r,t� = ur,pur,q� + iqu�,pur,q

r−

u�,pu�,qr

,

�A5�

bpq�r,t� = ur,pu�,q� + iqu�,pu�,q

r+

ur,pu�,qr

.

Defining semispectral energy as

Em�r,t� =12 �ur,mur,m

� + u�,mu�,m� � , �A6�

where � denotes complex conjugate, we can write

�Em�r,t��t

= Tm�r,t� = − �m=p+q

Tmpq = 0,

�A7�Tmpq =

12 �ur,m

� apq + u�,m� bpq + c.c.� .



Averaging in time and the radial direction

�Em�r,t� =2

T�b2 − a2��t0t1 �

a

b

Em�r,t�rdrdt , �A8�

where t0 and t1 are the starting and finishing times and T= t1− t0, we arrive at the desired result,

Pm =��Em�r,t�

�t= − �

m=p+q�Tmpq . �A9�

In the real experiment, this quantity would have been bal-anced by energy loss due to Ekman and viscous dampingand energy transfer due to mixed barotropic-baroclinicinteractions.

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32“Eddy” is defined here in the standard way as a deviation from the zonal�axisymmetric� flow profile. Thus any quantity f can be decomposed intozonal and eddy components: f�r ,� ,z�= f�r ,z�+ f�r ,� ,z��.


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